Question

if the slope of one of the lines represented by ax^+2hxy+by^=0 is three times the other then prove that 3h^=4ab
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Answer 1

Geometry

Given: the slope of one of the lines represented by $ax^{2}+2hxy+by^{2}=0$ is three times the other

To prove: $3h^{2}=4ab$

Proof:

Let $y^{2}+\frac{2h}{b}xy+\frac{a}{b}x^{2}=(y-m_{1}x)(y-m_{2}x)$

Then,

$\quad m_{1}+m_{2}=-\frac{2h}{b}$

$\quad m_{1}m_{2}=\frac{a}{b}$

Given, $m_{1}=3m_{2}\Rightarrow \frac{m_{1}}{m_{2}}=3$

Now, $\frac{(m_{1}+m_{2})^{2}}{m_{1}m_{2}}=\frac{(-2h/b)^{2}}{a/b}$

$\Rightarrow \frac{m_{1}^{2}+2m_{1}m_{2}+m_{2}^{2}}{m_{1}m_{2}}=\frac{4h^{2}/b^{2}}{a/b}$

$\Rightarrow \frac{m_{1}}{m_{2}}+2+\frac{1}{\frac{m_{1}}{m_{2}}}=\frac{4h^{2}}{ab}$

$\Rightarrow 3+2+\frac{1}{3}=\frac{4h^{2}}{ab}$

$\Rightarrow \frac{4h^{2}}{ab}=\frac{16}{3}$

$\Rightarrow\frac{h^{2}}{ab}=\frac{4}{3}$

$\Rightarrow 3h^{2}=4ab$

This completes the proof.